Is there a way to do this problem without a calculator

Determine whether the Mean Value Theorem applies to f(x) = 3x – x^2 on the interval [2, 3]. If it can be applied, find all value(s) of c in the interval that satisfy the Mean Value Theorem.

This is what I did

$\displaystyle \int f(x) dx$

$\displaystyle 3x^2/2 - x^3/3$

$\displaystyle F(3)=3*2^2/2-2^3/3=4.5$

$\displaystyle F(2)=3*2^2/2-2^3/3=3+1/3$

$\displaystyle 4.5-(3+1/3)=7/6$

$\displaystyle f(c)=3c-c^2$

I used my graphing calculator and found the intersection between f(c) and 7/6, and the answer is 2.5408.

Did I do it correctly? If I did, is there a way to do it without a calculator?